Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×#121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,23)(22,26)(24,27);(12,14,16,23,17,24,15,22)(13,18,19,21,20,26,25,27);(5,8)(6,11)(9,10);(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,21,26,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12089);